From OeisWiki

Pascal's triangle is a geometric arrangement of numbers produced recursively which generates the binomial coefficients.[1] It is named after the French mathematician Blaise Pascal (who studied it in the 17th century) in much of the Western world, although other mathematicians studied it centuries before him in Italy, India, Persia, and China. The triangle is thus known by other names, such as Tartaglia's triangle in Italy and much earlier (c. 500 BC) as the Yanghui triangle in China.

The rectangular version of Pascal's triangle(Figurate Number Triangle)[2]

= 0

1

1

1

1

2

1

2

1

3

1

3

3

1

4

1

4

6

4

1

5

1

5

10

10

5

1

6

1

6

15

20

15

6

1

7

1

7

21

35

35

21

7

1

8

1

8

28

56

70

56

28

8

1

9

1

9

36

84

126

126

84

36

9

1

10

1

10

45

120

210

252

210

120

45

10

1

11

1

11

55

165

330

462

462

330

165

55

11

1

12

1

12

66

220

495

792

924

792

495

220

66

12

1

= 0

1

2

3

4

5

6

7

8

9

10

11

12

In the equilateral version of Pascal's triangle, we start with a cell (row 0) initialized to 1 in a staggered array of empty (0) cells. We then recursively evaluate the cells as the sum of the two staggered above. The triangle thus grows into an equilateral triangle.

In the rectangular version of Pascal's triangle, we start with a cell (row 0) initialized to 1 in a regular array of empty (0) cells. We then recursively evaluate the cells as the sum of the one above left and the one directly above. The triangle thus grows into a rectangular triangle.

The outermost nonzero cells on each rows are therefore set to 1. All the interior cells are necessarily greater than or equal to 2 and the number of cells from rows 0 to which are equal to 1 is (Cf. A005408) and the number of cells from rows 0 to which are greater than or equal to 2 is , the thtriangular number.

See also

A003590 Rows written as a single base 10 number (the first five terms of that sequence match powers of 11; in general we can say that the first b/2 rows written as a single number give the powers of b + 1 in base b.

A006046 Total number of odd entries in first n rows of Pascal's triangle.

A003015 Numbers that appear five or more times in Pascal's triangle (at this point it's not known whether any terms appear exactly five times.)